Cahn–Hilliard Inpainting and a Generalization for Grayvalue Images

Burger, Martin; He, Lin; Schönlieb, Carola‐Bibiane

doi:10.1137/080728548

Cited by 145 publications

(158 citation statements)

References 31 publications

(56 reference statements)

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“…One would hope to achieve a graph-free stepsize rule as in the case of the original scheme without spectral truncation (8). However, as we show in our example below, a constant stepsize to guarantee monotonicity over all graph Laplacians of all sizes is not possible.…”

Section: A Counter Example For Graph-independent Stepsize Restrictionmentioning

confidence: 99%

“…We also note that the bound on dt depends linearly in , and we will generalize this dependency to include the fidelity term later in this section. To prove the proposition, we split the discretization (8) into two parts.…”

Section: Maximum Principle-l ∞ Estimatesmentioning

confidence: 99%

“…The most common among them is the Allen-Cahn equation [9], the L 2 gradient flow of the Ginzburg-Landau functional. Another commonly used model is the Cahn-Hilliard equation [2,8]. The diffuse interface models can often be used as a proxy for total variation (TV) minimization since the -limit of the Ginzburg-Landau functional is shown to be the TV semi-norm [20].…”

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confidence: 99%

“…Next, we introduce spectral truncation. Note in each iteration of (8) and (10), we need to solve a linear system of the form (I + dt L)u = v. In many applications, the number of nodes N on a graph is huge, and it is too costly to solve this equation directly. In [3,23], a strategy proposed was to project u onto the m eigenvectors of the graph Laplacian with the smallest eigenvalues.…”

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confidence: 99%

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Convergence of the Graph Allen–Cahn Scheme

Luo

Bertozzi

2017

J Stat Phys

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The graph Laplacian and the graph cut problem are closely related to Markov random fields, and have many applications in clustering and image segmentation. The diffuse interface model is widely used for modeling in material science, and can also be used as a proxy to total variation minimization. In Bertozzi and Flenner (Multiscale Model Simul 10(3):1090-1118, 2012), an algorithm was developed to generalize the diffuse interface model to graphs to solve the graph cut problem. This work analyzes the conditions for the graph diffuse interface algorithm to converge. Using techniques from numerical PDE and convex optimization, monotonicity in function value and convergence under an a posteriori condition are shown for a class of schemes under a graph-independent stepsize condition. We also generalize our results to incorporate spectral truncation, a common technique used to save computation cost, and also to the case of multiclass classification. Various numerical experiments are done to compare theoretical results with practical performance.

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Section: A Counter Example For Graph-independent Stepsize Restrictionmentioning

confidence: 99%

Section: Maximum Principle-l ∞ Estimatesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Convergence of the Graph Allen–Cahn Scheme

Luo

Bertozzi

2017

J Stat Phys

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“…IIP -Image Inpainting (IIP), a fourth-order partial differential equation called the Cahn-Hilliard Equation is solved numerically to propagate information smoothly from outside the data-missing region into it (Burger et al, 2009;Schönlieb, 2015). The inpainted image can be considered as a highly smoothness estimator of the original image and the "smoothness" was solved gradually until a stable state is reached.…”

Section: Gap Filling Methods and Artificial Gap Typementioning

confidence: 99%

A robust gap-filling method for Net Ecosystem Exchange based on Cahn–Hilliard inpainting

Rayment

2016

Preprint

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Abstract. Traditional gap-filling approaches adopt a temporally linear perspective on data; whether synthesizing data statistically within a moving window, or using complex functions based on a "best-guess" understanding of the processes driving exchange. The former approach is limited in its ability to capture non-linear trends, and the latter is limited in situations where the flux response to driving variables is poorly understood or unknown (e.g. the response of gas exchange to water table depth in wetlands). Rearranging time-averaged half-hourly net ecosystem exchange (NEE) into a 48*N matrix 10 has been used to visualize NEE as a "flux fingerprint" and suggests a different way of filling data gaps. In this paper, we introduce an image processing technique known as image inpainting to fill gaps in this two-dimensional representation of a one-dimensional data. This has the advantage that any short-term structure can be accommodated without expressly implying any particular functional response to driving environmental variables, and medium-term temporal structure (i.e. day-to-day covariance) can be incorporated into gaps in the flux signal. In this way, data gaps are filled solely using information 15 contained in robust, primary data. This new method compares favorably with the marginal distribution sampling (MDS), when tested on twelve European-Flux datasets with four types of artificial gaps. Furthermore, we show that how random structures or noise embedded in the signal affect the gap-filling performance, which can simply be improved through a denoising procedure by using a Fourier transform algorithm. The inpainting-based gap-filling approach is more effective than MDS on the de-noised data. 20

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A multiscale fourth‐order model for the image inpainting and low‐dimensional sets recovery

Theljani

Belhachmi

Kallel

et al. 2016

Math Methods in App Sciences

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We consider a fourth‐order variational model, to solve the image inpainting problem, with the emphasis on the recovery of low‐dimensional sets (edges and corners) and the curvature of the edges. The model permits also to perform simultaneously the restoration (filtering) of the initial image where this one is available. The multiscale character of the model follows from an adaptive selection of the diffusion parameters that allows us to optimize the regularization effects in the neighborhoods of the small features that we aim to preserve. In addition, because the model is based on the high‐order derivatives, it favors naturally the accurate capture of the curvature of the edges, hence to balance the tasks of obtaining long curved edges and the obtention of short edges, tip points, and corners. We analyze the method in the framework of the calculus of variations and the Γ‐convergence to show that it leads to a convergent algorithm. In particular, we obtain a simple discrete numerical method based on a standard mixed‐finite elements with well‐established approximation properties. We compare the method to the Cahn–Hilliard model for the inpainting, and we present several numerical examples to show its performances. Copyright © 2016 John Wiley & Sons, Ltd.

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Cahn–Hilliard Inpainting and a Generalization for Grayvalue Images

Cited by 145 publications

References 31 publications

Convergence of the Graph Allen–Cahn Scheme

Convergence of the Graph Allen–Cahn Scheme

A robust gap-filling method for Net Ecosystem Exchange based on Cahn–Hilliard inpainting

A multiscale fourth‐order model for the image inpainting and low‐dimensional sets recovery

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